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CONICAL CATENARY
Curve studied by Bobillier in 1829. 
The conical catenary is the equilibrium line of an inelastic flexible homogeneous infinitely thin massive wire included in a cone of revolution, placed in a uniform gravitational field.
Differential equation:
(see at general catenary),
where
is the normal vector of the cone.
Case of the vertical cone with vertex O and halfangle a, parametrization based on the polar coordinates of the development plane: .

(but that is not a hyperbola!) 
Differential equation in the case where :
Parametrization: (rectangular hyperbola in the development plane). 


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© Robert FERRÉOL,
Alain ESCULIER 2018